What are some of the advantages in adding Spring into a Flex / AIR application? And When would you recommend using Spring?
Flex is addressing what happens in the UI. You are likely to need some server-side logic to provide the data for the UI. Writing robust, performant, secure server-side logic from scratch is not trivial. Frameworks such as Spring, EJB 3, etc. really do help with that. The learning curve for modern frameworks is not too steep, getting started is barely more that a few extra "annotations" to the code you knew you wanted to write.
Well worth investigating.
The sum total of learning Flex + Spring + also some DB admin is still quite a lot. You may need to keep your ambitions under control, don't try to fly before you are walking comfortably.
Good luck.
If you're writing a Flex app with a Java backend, you'll want to use BlazeDS to handle remoting and/or messaging between the client and server. Spring has support for integrating with BlazeDS and making Blaze a little easier to setup and configure. It's also pretty easy to set up BlazeDS on its own, so if you don't want to learn Spring then you should not be afraid to use BlazeDS by itself.
The bulk of what Spring provides is specific to the Java layer but it will help you write better Java applications. In particular it provides a clean way for wiring up application components, simplifies data access using Hibernate and/or JDBC, makes it easy to do aspect-oriented programming and also has a pretty nice security framework (Spring Security). There's a ton that Spring can do and I recommend using Spring for pretty much any Java backend.